On the cohomology of a general fiber of a polynomial map

C ^OMPOSITIO M ^ATHEMATICA

A ^LEXANDRU D ^IMCA M ^ORIHIKO S ^AITO

On the cohomology of a general fiber of a polynomial map

Compositio Mathematica, tome 85, n^o3 (1993), p. 299-309

<http://www.numdam.org/item?id=CM_1993__85_3_299_0>

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http://www.numdam.org/

On the

cohomology

^{of a}

general

^{fiber of a}

polynomial

map

ALEXANDRU DIMCA1 and MORIHIKO SAITO2

’School of Mathematics, The University of Sydney, Sydney ^NSW 2006, Australia; 2RIMS Kyoto University, Kyoto ⁶⁰⁶ Japan

Received 11 October 1991; accepted ²¹ January ¹⁹⁹² (Ç)

Introduction

Let X ⁼ C", ^{S =} C, ^and f : ^{X - S a map defined} by ^a polynomial which is also denoted by f. ^Then f ^{induces a} topological ^{fibration over a} Zariski-open ^subset

of S. It is interesting ^{whether we can} compute algebraically ^the cohomology ^{of a} generic ^{fiber F =}

f-1(t)

using ^the polynomial ring (9: = C[x1,..., xn] ^{and the} polynomial f ^Of course, ^{we can} calculate the cohomology using ^{the de Rham} cohomology ^{of the} (scheme theoretic) generic ^{fiber of} f, but it is not quite computable.

In the weighted homogeneous ^{case, the} answer was given by [3]. ^{Let Q’}

denote the complex ^of global algebraic differential forms on X (i.e., ^{03A9p is a free U-}

module with a basis dxi ^ ··· ^A

dxip

^{(il ...}

ip)).

^{Define a} differential D f ^on

03A9 by

Then we have an isomorphism

if f ^is weighted homogeneous, ^where H denotes reduced cohomology. (In [loc. cit.], D f ^{was denoted by}

Df.

^{See also} (2.10) below.) ^{In this} paper, ^we prove

(0.3) THEOREM. The isomorphism (0.2) holds for any polynomial f.

The proof ^{uses the} theory ^of algebraic Gauss-Manin system ^{which is a} generalization ^of [4] (see, ^for example, [1]), and also the theory of monodrom- ical algebraic D-modules.

Let ~fOX

denote the algebraic Gauss-Manin system, which is defined by the direct image ^of (9x by f ^as algebraic D-module. Let t be the coordinate of S, ^and ôr ⁼ olot. ^{Then we have a natural} quasi-isomorphism

(see (2.7)):

Here ôt - id ^is analytically equivalent ^to ô, (because ~t-1=et ôt e - ⁱⁿ DSan). ^Let

~pfOX

denote the pth cohomology

of f f

^{(9x. Its} restriction to a Zariski open subset of S is a vector bundle (i.e., ^a locally ^free sheaf) whose fiber is isomorphic ^{to the} cohomology ^of the fiber of f (see (2.3)). ^{We take the direct} image

of IP

^{(9x by the}

compactification ^{S -} P 1, ^and compute ^its analytic ^local cohomology ^at infinity (see (2.8)). ^{Then we} get ^{the assertion} using ^the theory ^of monodromical D- modules (see (2.9)).

It should be noted that Theorem (0.3) ^is essentially of algebraic nature, and the local analytic ^version of (0.3) ^{does not hold. For} example, (03A9Xan,

Df)

[1] ^{is not} quasi-isomorphic ^to Deligne’s vanishing cycle ^{sheaf, because} D f ^is analytically equivalent ^to the natural differential d using ^{ef .}

1. Monodromical D-modules of ^one variable

In this section, ^we gather ^some elementary facts from the theory of monodrom- ical algebraic -9-modules of one variable, which should be well known to

specialists.

(1.1) Let S denote the affine line C with coordinate t (i.e., ^{S =} Spec C[t]). ^Let

S* ⁼

SB{0}

^{with a natural} inclusion j : S* ~ S. Let DS be the sheaf of algebraic

differential operators ^{on S} [1], [5]. ^{We denote} by R ^the global sections of -9s,

which is the Weyl algebra C[t, ôt]. ^Let Mcoh(DS) ^{be the} category of coherent -9s- modules, ^and Mfi.(R) ^the category of finite R-modules. We have an equivalence

of categories

by ^the global ^{section functor} r(S, *).

Let Sa" denote the underlying complex analytic space of S. We have a functor

by

M ~ Man := OSan ~OS M,

^{where the} pull-back by ^{the natural} morphism

Sa" ~ S is omitted. Then the de Rham functor DRS ^is given by

using the coordinate t to trivialize

03A91S

(see (2.1.2) below).

(1.2) ^{For M E} Mcoh(DS), ^let M(S) ⁼ r(S, M), ^and

Then

and we have isomorphisms

In fact, tôt ^is bijective ^on M(S)" for 03B1 ~ 0, ^because tôt ^{= a on}

GrKM(S)"

^with KiM(S)" ⁼

Ker(t~t-03B1)i+1

(similarly ^for a,t).

(1.3) DEFINITION. We _say that M~Mcoh(DS) is monodromical if M is generated by M(S)" (03B1~C) ^over -qs. ^Let Mcoh(DS)mon ^{denote the full sub-} category ^of Mcoh(DS) consisting ^of monodromical DS-modules. ^Then

M E Mcoh(DS)mon is ^called meromorphic (resp. microlocal) type if the action of t

(resp. ôt) ^on M(S) ^is bijective.

REMARK. The condition of monodromical DS-module ^is equivalent ^{to that}

any element of M(S) is annihilated by ^a polynomial ^of tôt. ^{So it is stable} by

extensions in M coh( !?fis).

(1.4) ^{LEMMA. For M E} Mcoh(DS)mon, ^{we have a natural} isomorphism

and M(S)" ^is finite dimensional over C. In particular, ^the functor ^{M ~} M(S)03B1 ^is

exact.

Proof. ^The injectivity ^of (1.4.1) ^{is clear} using ^{the action} of tôt ^on M(S). ^{Since the}

condition of monodromical DS-module ^is equivalent ^{to the} surjectivity ^of

the surjectivity ^of (1.4.1) follows from (1.2.3), taking ^the global section of (1.4.2).

We have dimcM(S)a ^{oo, because} M(S)" ^is finitely generated ^over C[N] ^with N = - (tôt - a).

REMARK. By (1.2.3) ^and (1.4.1), M~Mcoh(DS)mon ^is meromorphic (resp.

microlocal) type ^{if and} only ^if

is bijective.

(1.5) ^{LEMMA. Let M} ~Mcoh(DS) such that supp M ^c

{0}.

^{Then M is monodrom-}

ical, ^and M(S)" ^{= 0} except for negative integers ^a.

Proof. ^The assumption ^is equivalent ^to that any element of M(S) is annihilated

by ^a sufficiently high power of t. Then we can check the assertion using

~itti

⁼

03A00ji

^{(tat +j).}

REMARK. For M as above, ^{M is a finite direct sum of in the} proof of (1.8) by (1.2.3). ^{This is a} special ^case of Kashiwara’s equivalence (see [1]).

(1.6) LEMMA. Let A be a subset of ^C such that 0 E A and the natural morphism

A -+ C/Z ^is bijective. ^{Let A’ =}

039B~{-1}.

Let CC be the category ^whose object ^{is a} famil y of C-vector spaces V03B1(03B1 ^E A’) ^with morphisms ^{u : V0 ~} V-1, ^{v : V-1 ~} VO, ^and

N: V03B1 ~ V03B1 (a ^E

039BB{0})

^{such that} ~03B1~039B’ ^{V" is} finite dimensional, ^and vu, ^uv and N

are nilpotent. ^{Then we have an} equivalence of categories

by associating M(S)", ~t, ^{t and} tô, ^{- a to M E} Mcoh(DS)mon.

Proof. This follows from (1.2.2-3) ^and (1.4.1).

(1.7) COROLLARY. We have an equivalence of categories

where the left-hand ^{side is the} category of monodromical -9s-modules of ^mero- morphic (resp. microlocal) type, ^the right-hand ^{side is the} category of finite

dimensional C-vector _spaces with an automorphism T, ^and the functor ^is defined by

M - ~03B1~039B M(S)" ^{with T =} exp( - 2nitô,).

REMARK. Using (1.6), ^{we can} show that the category Mcoh(DS)mon ^is equivalent

to the category of regular ^holonomic

DSan,0-modules

(for ^{which an} equivalence ^of categories ^{similar to} (1.6.1) holds). ^{The terms} ’meromorphic’ ^and ’microlocal’ are

originally ^{used in this case} (see [8]).

(1.8) PROPOSITION. Let M~Mcoh(DS)mon. ^{Then M is} regular ^holonomic [1].

Proof. ^{Since the} action of t~t-03B1 ^on M(S)" ^is nilpotent, ^we may ^assume

03A303B1~039B’ ^dim M(S)" - ¹ by (1.6), taking ^the graduation ^{of a finite} filtration on M

(because regular ^{holonomic D-modules are stable} by extensions [loc. cit.]). ^Then

we can check that M is isomorphic ^{to one of the} following:

depending ^{on the a such that} M(S)03B1 ~ ^{0. So we} get the assertion.

REMARK. We can show that a regular ^holonomic DS-module is monodrom-

ical, ^{if and} only if its restriction to S* is finite over Us* (i.e., ^{a vector} bundle with connection [2]). ^{In fact, we may assume that} M|S* ^{is a vector bundle} by (1.10)

below. Then the assertion is reduced to case where the action of t on M is

bijective using ^the localization of M by ^t (because Mcoh(DS)mon ^{is stable} by

extensions in M,.,(-9s), ^{see Remark after} (1.3)). ^{Then the} assertion follows [2]

(see ^also (1.11) below).

(1.9) PROPOSITION. For M~Mcoh(DS)mon, ^{there exists} uniquely

M’ E Mcoh(DS)mon of meromorphic (resp. micro local) ^{type with a} morphism ^{M ~ M’}

(resp. ^{M’ ~} M) inducing ^an isomorphism ^{on S*.}

Proof. By (1.6) there exists uniquely M’~Mcoh(DS)mon ^{with a} morphism

M ~ M’ (resp. ^{M’ ~} M), ^{such that}

for i &#x3E; 0. Then the morphism ^{induces an} isomorphism ^{on S*} by (1.5).

REMARK. In the standard notation (see [1]), ^{M’ is denoted} by

j* j*M

(resp. j! j*M). Here j* ^is really ^{the direct} image ^{as Zariski sheaf} (because ^{M’ is the}

localization of M by t), but j! ^{is not. In} factjt ^{is defined} by Dj*D ^{with D the dual}

functor (see [loc. cit.]). ^{We have}

See also (1.12) ^{below for} (1.9.3).

(1.10) COROLLARY. For M~Mcoh(DS)mon, ^the restriction of ^{M to S* is} a free OS*-module of ^rank 1,,c-A ^dim M(s)a. ^In particular, Man|S* ^{is a vector} bundle with connection [2], ^and DRS(M)[-1] Is. ^{is a} local system.

Proof. ^{It is} enough ^to show the first assertion. We may ^{assume M mero-}

morphic type by (1.9). ^Then M(S) ^{is a free} C[t,t-1]-module ^of

rank I:aeA ^dim M(S)03B1 by (1.2.3) ^and (1.4.3), ^{and the} assertion follows.

(1.11) PROPOSITION. Let L be a local system ^{on S*} with complex coefficients, L~ ^the group of multivalued sections of ^{L with the} monodromy ^{T, and} L03B1~ ^the

exp(-203C0i03B1)-eigenspace of Loo with respect ^to 1;, ^{where T =} TsTu ^{is the Jordan} decomposition. Then there exists uniquely ^{M E} Mcoh(DS)mon of meromorphic (resp.

microlocal) type with ^an isomorphism

where DRs ^{is as in} (1.1.3). Furthermore, ^{we have a canonical} isomorphism

for a E A, ^such that - (tôt - a) corresponds ^{to N: =} (log Tu)/203C0i.

Proof. By (1.9) ^{it is} enough ^{to show the assertion for M} meromorphic type. By (1.7), ^{there exists} uniquely M~Mcoh(DS)mon ^of meromorphic type ^{with the}

isomorphism (l.11.2). ^{Then Man is} identified with a

OS[t-1]-submodule

^of j*«(9s* ~CL) generated by

for u e Li with a E A (see [2]), ^because (1.11.3) ^{satisfies the same relation as the}

element of M(S)03B1 corresponding ^{to u E} Li by definition of M (and ^Man|S* ^and (!Js* Q9c L ^{have the same} rank). ^In particular, Man|S* = OS* ~C L, ^{and the}

assertion follows.

REMARK. This proposition ^{shows that we have an} equivalence ^of categories

between the category of monodromical DS-modules ^of meromorphic type ^and the category ^{of local} systems ^on S*, ^{in a} compatible way with (1.7.1). ^{Note that}

the isomorphism (1.11.2) depends ^on the choice of the branch of log ^t (i.e., ^the

choice of a lift of 1 to a universal covering ^of S*).

Proof. By (1.10), DRS(M)[-1]|S* ^{is a local} system, and it is enough ^{to show} (1.12.1). Using ^a filtration defined in the category of monodromical éos-modules

of microlocal type, ^we may ^assume 03A303B1~039B ^dim M(S)03B1 = 1. Then it is isomorphic ^to M(03B1)=DS/DS(t~t-03B1) if M(S)03B1 = C ^for

03B1~039BB{0}

^{(see the} proof ^of (1.8)). ^{In the}

other _case, we can check that M is isomorphic ^to DS/DS(t~t). ^{Then we can check}

the assertion (see ^also [8]).

2. Algebraic Gauss-Manin system

(2.1) ^Let f:X ~ ^{Y be a} morphism ^{of smooth} complex algebraic varieties. The direct

image f f

^{M of a DX-module M is defined by}

where _03C9X is the dualizing ^{sheaf, and "} denotes the dual line bundle. See [1], [6], [10], ^{etc. Note that, if} f ^{is an open} embedding,

If M

^{is defined by the sheaf}

theoretic direct image. ^{In the case Y} is the affine line S, the direct

image If

^{M will}

be more explicitly expressed ^later (see (2.6)).

We denote the cohomological ^direct image

JfP Jf

^M

by Jj

^{M. For M = (9,, the}

direct

image ~f OX

^(or

~pfOX)

is called the Gauss-Manin system ^of f [7].

For a DX-module ^{M, let}

where 03A9X(M) denotes the de Rham complex ^{as in} [1], [2], ^{and an is defined as in} (1.1.2). By [1] ^{we have:}

(2.2) PROPOSITION. If M ^is regular holonomic, ^the cohomological ^direct images

~pf

^{M are regular} holonomic, ^{and we have a natural} isomorphism

(2.3) COROLLARY. Assume f smooth with relative dimension r, ^and

~pf

^{OX is a}

locally free OY-module of finite ^rank (i.e., a ^vector bundle with integrable ^connection [2]). ^{Let LP} be the local system defined by the horizontal sections of

(~pf

^{OX)an. Then}

we have natural isomorphisms

Proof. ^The first assertion follows from the Poincaré lemma. Then the second follows from (2.2.1) using DRX(OX) = CXan [dim X], ^because (9x ^is regular

holonomic by definition [1].

REMARK. If we assume that

RP+1*Cxan

^{is a local system, the} corollary ^follows

also from the relative version of [4] using ^a desingularization ^{of the divisor at} infinity ^{of a} compactification ^of f, ^{because we may} replace ^Y by ^its Zariski-open

subset.

(2.4) ^LEMMA. Let f : ^{X ~ Y be as in} (2.1), ^{and assume Y is the} affine line S with coordinate t as in Section 1 so that f ^is identified ^{with a} function ^{on X. Let}

0 ⁼ at-id, ^and

Then we have a natural isomorphism

where an is defined ^{as in} (1.1.2).

Proof. This follows from ô, ^{-1 =} etot ^{e - in} DanS.

(2.5) PROPOSITION. For f : ^{X ~ S as} above, ^{assume Xan} contractible and purely

n-dimensional. Then we have a natural isomorphism

Proof. This follows from (2.2) ^and (2.4).

REMARK. We can apply (2:2) ^{also to} the direct image ^of (9x by X ~ pt ^{and the}

direct image

of ~pf

(9x by S - pt. ^{In this} case, (2.2) ^{means the} commutativity ^{of the}

direct image ^{with the} functor An in (1.1.2), ^and follows also from [4] ^and [2] (see

also (1.9.2) above) respectively.

(2.6) ^Let f : X ~ S ^{be as in} (2.4), ^{and M a} -9x-module. ^{We define a structure of} éox-module ^{on M} Q9c C[~t] by

for g~OX, 03B6~0398X ^{and U E M.} It has also the action of R ⁼ C[t, ôt] (see (1.1)) by

which commutes with the action of -9,. ^{Then M} ~C[~t] is identified with the direct image ^{of M} by ^the

embedding i f

by ^the graph of f, ^{and u}

Q9 G:

is ^identified

with

~it03B4(t -

f ) Q ^{u. Here} 03B4(t - f ) is the delta function with support

f f

⁼

tl,

^and

satisfies the relation

which gives (2.6.1-2).

Since the direct image ^{of a -9-module} by ^{a smooth} projection ^{with fiber X is} given by ^the sheaf theoretic direct image ^{of the relative de Rham} complex ^shifted by ^{dim X} (see ^for example [1]), the direct

image ~f M

^{is expressed as}

factorizing f ^into the closed

embedding i f

^{and thé} projection. ^{This can be also}

obtained by using ^{induced D-modules} [9]. ^Note that, if f is ^{an affine} morphism,

the derived direct image

Rf,

^{can be} replaced by f*.

(2.7) PROPOSITION. For f : X ~ S ^as above, ^{assume X =} C". Then we have a natural quasi-isomorphism (0.4) ⁱⁿ the introduction.

Proof. By (2.6.4),

~fOX

^is expressed by

f*(nx

~CC[~t])[n], ^{where the}

differential of 03A9:X Oc C[ô,] ^is given by

See (2.6.1). ^{Then we have a short exact} sequence of complexes

where the differential of Qx ^{is D} f . Then we get the assertion taking ^{the exact}

functors f* ^and F(S, *).

(2.8) PROPOSITION. Let S ⁼ pl with a natural inclusion j’ : ^{S ~ S. Let M be a} regular ^holonomic DS-module, ^{and K} = Cone(03B8: M ~ M) for ^{0 as} above. Let U be a

Zariski-open ^subset of ^{S on which M is} locally free ^over OU, ^{and denote the local}

system DRS(M)[-1] luan by ^{L. Then we have a canonical} isomorphism

for ^t~Uan(^c C) ^{such that Im t =} 0 and Re t » 0. Furthermore,

Proof. ^Let

V = {t~C:|t|&#x3E;R} ~ Uan

^{for R} sufficiently large, ^{and V’ =}

V ~ {~}.

^Then (j’M)" 1 v, ^{is an} extension of Man|V ^as regular ^holonomic DV’-

module such that the action of the local coordinate

s(=t-1)

^is bijective, ^and

such an extension is unique by [2]. ^So

Hi{~}((j’*K)an)

^{is uniquely} determined by Man|V, ^or equivalently, by L|V (see [loc. cit.]). Since V is homotopy equivalent ^to (s*)an, ^we may ^assume U = S*, ^{and M is a} monodromical DS-module ^of

microlocal type by (1.11). ^Then

by ^{the same} argument ^{as the} proof of (2.4). ^{So it is zero} for any ⁱ by (1.12.2), ^and

we may replace

Hi{~}((j’*K)an)

ⁱⁿ (2.8.1-2) by

Hi(San,

(j’*K)an) using ^the long ^exact

séquence :

By GAGA, ^{we have}

where the last isomorphism ^{follows from the exactness} of j’. ^Moreover,

using (1.2.3) ^and (1.4.3). ^So the assertion follows from the isomorphism (1.11.2).

Here we identify L, ^with L~ by taking ^{a lift of t to a universal} covering ^of S*, ^at

which log t is real valued.

(2.9) Proof of ^Theorem (0.3). ^Let

K f

^{be as in (2.4), and j’ : S ~ S as above. By}

GAGA and (2.7), ^{we have}

We have the long ^exact sequence (2.8.4) ^{with K} replaced by K f. ^{Let F =}

f -1(t)

for t as in (2.8.1). ^{Then it is} enough ^{to show a canonical} isomorphism

by (2.5.1), ^{because we can check}

H0(03A9·, Df)

^{= 0 so that the morphism}

is injective. ^Then, applying (2.8) ^{to M}

= ~pf

^{OX, the assertion (2.9.2)} follows from

(2.3).

(2.10) ^REMARKS. (i) ^If f ^is weighted homogeneous,

~pfOX

^{(p * 1-n) and}

~1-nf OX/OS

^are monodromical DS-modules of microlocal type, ^{and the decom-} position (1.4.1) by the action of t~t is induced by ^the grading ^{of Q’} compatible

with f ^{as in} [3]. ^This implies ^{that the} isomorphism (0.2) ^is compatible ^{with the}

action of monodromy ^{as in} [loc. cit.].

(ii) ^{In Theorem A of} [3], b ^{does not induce an} isomorphism for k = 0. The definition

of D f

^{should be} replaced by (0.1) in this paper, which is denoted by

D f

in [loc. cit.].

(iii) ^{In the} proof of ( 1.8) of [3], it is better to use Coim 0394 instead of 03A9 = Ker A.

(2.11) ^EXAMPLE: f ⁼

x2y+x.

^{This is a} generalized weighted homogeneous polynomial admitting negative weights, ^{and the} spectral sequence ^as in [3] ^does

not converge. In fact, ^the Eo-complex ^is isomorphic ^to the Koszul complex (QB d f 039B), ^{and is} acyclic, ^{but the} general ^{fiber F =}

f -1(t)

^is isomorphic ^{to C* so}

that

H°(F,

C) ⁼ 0,

H1(F,

C) ^{= C. We can check}

H2(03A9, Df)

^{= C as follows.}

We have fx ⁼ 2xy +

1, fy

^{= x2, and}

Let ~:03A92 ~ C

^{be a} map defined by

~(xiyjdxdy)=(-1)ij!/(2j-i+1)!

^for 2j +1 i, ^{and 0} otherwise. Then 0 ^{induces the} isomorphism

H2(03A9·, Df)

^{= C.}

Added in the proof. ^{We are} informed that a similar result is obtained by ^B.

Malgrange ^{and P.} Deligne independently using ^the theory of Fourier transformation.

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